Topology of the immediate snapshot complexes

The immediate snapshot complexes were introduced as combinatorial models for the protocol complexes in the context of theoretical distributed computing. In the previous work we have developed a formal language of witness structures in order to define and to analyze these complexes. In this paper, we study topology of immediate snapshot complexes. It is known that these complexes are always pure and that they are pseudomanifolds. Here we prove two further independent topological properties. First, we show that immediate snapshot complexes are collapsible. Second, we show that these complexes are homeomorphic to closed balls. Specifically, given any immediate snapshot complex P(r̄), we show that there exists a homeomorphism φ : ∆|supp r̄|−1 → P(r̄), such that φ(σ) is a subcomplex of P(r̄), whenever σ is a simplex in the simplicial complex ∆|supp r̄|−1. 1. Witness structures and immediate snapshot protocol complexes 1.1. Modeling protocol complexes for the immediate snapshot read/write distributed protocols. A crucial ingredient in the topological approach to theoretical distributed computing, see Herlihy et al, [HKR], is associating a simplicial complex, called the protocol complex, to every distributed protocol, once the computational model is fixed. In this paper, we study topology of standard full-information protocol complexes in one of the central models of computation. Let us fix the computational model to be the immediate snapshot read/write model, which was originally introduced by Borowsky and Gafni in [BG]. Roughly, this means that the processes can write their values to the assigned memory registers, and they can read the entire memory in one atomic step (snapshot read). The execution of the protocol must have a layer structure, where in each layer a group of processes becomes active, the processes in this group atomically write their values to the memory, after this they atomically read the entire memory. Importantly, there are no further restrictions on how these layers get activated during the protocol execution. In our previous work, [Ko14b], we introduced combinatorial models for the protocol complexes for the standard protocols in that chosen computational model, called immediate snapshot complexes. For this, we needed to define new combinatorial structures, called witness structures, and study their structure theory, including various operations, such as ghosting. We have proved that the immediate snapshot complexes provide the correct model for these protocol complexes, and started to study their topology. The standard protocols are naturally enumerated by finite sequences of nonnegative integers, which we called round counters, denoted r̄. Accordingly, the immediate snapshot complexes themselves were denoted P(r̄). In [Ko14b] it was proved that the complexes